1 Load diagram on beam

2 Beam diagram

3 Shear diagram

4 Bending diagram

The area method may be stated:

• The shear at any point n is equal to the shear at point m plus the area of the load diagram between m and n.

• The bending moment at any point n is equal to the moment at point m plus the area of the shear diagram between m and n.

The shear force is derived using vertical equilibrium:

∑ V = 0; Vm - w x - Vn = 0; solving for Vn

Vn = Vm-wx

where w x is the load area between m and n (downward load w = negative).

The bending moment is derived using moment equilibrium:

∑ M = 0; Mm + Vmx – w x x/2 - Mn = 0; solving for Mn

where Vmx – wx^2/2 is the shear area between m and n, namely, the rectangle Vm x less

__the triangle w x^2/2. This relationship may also be stated as Mn = Mm + Vx, where V is the average shear between m and n.__

By the area method moments are usually equal to the area of one or more rectangles and/or triangles. It is best to first compute and draw the shear diagram and then compute the moments as the area of the shear diagram.

From the diagrams and derivation we may conclude:

• Positive shear implies increasing bending moment.

• Zero shear (change from + to -) implies peak bending moment (useful to locate maximum bending moment).

• Negative shear implies decreasing bending moment. Even though the forgoing is for uniform load, it applies to concentrated load and non-uniform load as well. The derivation for such loads is similar.

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